Endomorphisms of the semigroup of order-preserving mappings.
type: Item
instance of: scholarly article
title: Endomorphisms of the semigroup of order-preserving mappings.
zbMATH Open document ID: 1207.20059
published in: Semigroup Forum
publication date: 2010-10-27T00:00:00Z
review text: An endomorphism \(f\) of \(\{1,2,\dots,n\}\) is called order preserving provided that \(i\leq j\) implies \(f(i)\leq f(j)\). The main result of this paper gives a complete classification of the semigroup of all order-preserving endomorphisms of \(\{1,2,\dots,n\}\). It turns out that every endomorphism \(\varphi\) of this semigroup has one of the following three types: (i) \(\varphi\) is an automorphism (and hence is either the identity or the conjugation by the order reversing permutation); (ii) \(\varphi\) is constant (and hence maps the whole semigroup to an idempotent); (iii) the image of \(\varphi\) contains exactly two elements, both these elements are idempotents, one of them is the image of the identity (which coincides with the group of invertible elements) and the other one is the image of the singular part.
reviewed by: Volodymyr Mazorchuk
Mathematics Subject Classification ID: 20M20
Mathematics Subject Classification ID: 20M15
zbMATH DE Number: 5806755
zbMATH Keywords: semigroups of order-preserving mappings
zbMATH Keywords: endomorphisms
zbMATH Keywords: transformation semigroups
zbMATH Keywords: automorphisms
zbMATH Keywords: idempotents
author: Vítor H. Fernandes
author: Manuel M. Jesus
author: Victor Maltcev
author: J. D. Mitchell
MaRDI profile type: Publication
full work available at URL: s00233-010-9220-7
OpenAlex ID: W2072728273
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DOI: 10.1007/S00233-010-9220-7
description: scientific article; zbMATH DE number 5806755